528 Computational methods for lattice field theories
these theories and QED is that the commuting complex phase factorsUμ(n)of
QED are replaced by noncommuting matrices, members of the group SU(2) (for
the weak interaction) or SU(3) (strong interaction). Furthermore, in quantum chro-
modynamics (QCD), the SU(3) gauge theory for strong interactions, more than
one fermion flavour must be included. In this section we focus on QCD, where the
fermions are thequarks, the building blocks of mesons and hadrons, held together
by the gauge particles, calledgluons. The latter are the QCD analogue of photons
in QED.
Quarks occur in different species, or ‘flavours’ (‘up’, ‘down’ ‘strange’...); for
each species we need a fermion field. In addition to the flavour quantum number,
each quark carries an additional colour degree of freedom: red, green or blue.
Quarks form triplets of the three colours (hadrons, such as protons and neutrons),
or doublets consisting of colour–anticolour (mesons): they are always observed in
colourless combinations. Quarks can change colour through the so-calledstrong
interactions. The gluons are the intermediary particles of these interactions. They
are described by a gauge field of the SU(3) group (see below). The gluons are
massless, just as the photons in QED.
The U(1) variables of QED were parametrised by a single compact variableθ
(U =exp(iθ)). In QCD these variables are replaced by SU(3) matrices. These
matrices are parametrised by eight numbers corresponding to eight gluon fields
Aaμ,a=1,..., 8 (gluons are insensitive to flavour). The gluons are massless, just
like the photons, because inclusion of a mass termm^2 AμAμ, analogous to that of
the scalar field, destroys the required gauge invariance.
Experimentally, quarks are found to have almost no interaction at short separa-
tion, but when the quarks are pulled apart their interaction energy becomes linear
with the separation, so that it is impossible to isolate one quark. The colour interac-
tion carried by the gluon fields is held responsible for this behaviour. There exists
furthermore an intermediate regime, where the interaction is Coulomb-like.
The fact that the interaction vanishes at short distances is called ‘asymptotic
freedom’. It is possible to analyse the behaviour of quarks and gluons in the short-
distance/small coupling limit by perturbation theory, which does indeed predict
asymptoticfreedom(G.’tHooft,unpublishedremarks, 1972 ;andRefs.[ 55 , 56 ]).
The renormalised coupling constant increases with increasing distance, and it is this
coupling constant which is used as the perturbative parameter. At length scales of
about 1 fm the coupling constant becomes too large, and the perturbative expansion
breaks down. This is the scale of hadron physics. The breakdown of perturbation
theory is the reason that people want to study SU(3) gauge field theory on a com-
puter, as this allows for a nonperturbative treatment of the quantum field theory.
The lattice formulation has an additional advantage. If we want to study the time
evolution of a hadron, we should specify the hadron state as the initial state. But